Principles of Superconducting Quantum Computers by Stancil Daniel D.;Byrd Gregory T.; & Gregory T. Byrd

Author:Stancil, Daniel D.;Byrd, Gregory T.; & Gregory T. Byrd [D. Stancil, Daniel & Byrd, Gregory T.]
Language: eng
Format: epub
Publisher: John Wiley & Sons, Incorporated
Published: 2022-04-08T00:00:00+00:00

7.4.2 Dielectric Function

When an electric field is applied to a material, charges are redistributed and/or induced in such a way that the electric field generated by the new charge distribution opposes the applied electric field. As a result, the effect of the applied field is diminished or “screened.” The dielectric function, or permittivity, describes the screened response of the material. As an example, let us take the interaction energy between the Cooper pair electrons to be the Coulomb interaction

(7.69)

For two isolated electrons in a vacuum, the dielectric function is simply the permittivity of free space, i.e., . However, in a physical material, the electrons and ions can redistribute themselves in such a way as to screen the two electrons from one another. In this case, . However, depending on the dynamics of the particles in the material, the dielectric function can be less than that of free space, or even negative. The case of a negative dielectric function is of particular interest to us, since this would turn the repulsive Coulomb potential (7.69) into an attractive potential—just what we need to form Cooper pairs.

We therefore need to develop a model for the dielectric function. To do this it will be helpful to obtain a relation between the dielectric function and the imposed and induced charges. Let us assume that there is no net charge in the medium in the absence of an imposed perturbation. From Gauss’ law we have

(7.70)

where δρ is the imposed charge fluctuation. However, we also know that . Substituting this expression for D gives

(7.71)

where the total charge density is equal to the imposed charge density δρ plus the screening response of the medium .

Since we are interested in phonon scattering, let us consider the case where δρ and are proportional to :

(7.72)

We define the dielectric function by the constitutive relation . This enables us to write

(7.73)

However, from (7.70). Making this substitution and solving for ε gives the relation we are looking for between the imposed and induced charge densities and the dielectric function:

(7.74)

The screening charge density is composed of contributions from both the movement of the electrons as well as the background ions

(7.75)

In the next section we will examine simple models for both contributions.

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